Takashi Shioya

Fixed point sets of parabolic isometries of CAT(0)-spaces

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most

On Generalized Measure Contraction Property and Energy Functionals over Lipschitz Maps

\pi/2

THE LIMIT SPACES OF TWO-DIMENSIONAL MANIFOLDS WITH UNIFORMLY BOUNDED INTEGRAL CURVATURE

, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

Variational convergence over metric spaces

We construct Sobolev spaces and energy functionals over maps between metric spaces under the strong measure contraction property of Bishop–Gromov type, which is a generalized notion of Ricci curvature bounded below. We also present the notion of generalized measure contraction property, which gives a characterization of energies by approximating energies of Sturm type over Lipschitz maps.

Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

We study the class of closed 2-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally 2-connected. We also study the limit of 2-manifolds with Lp-curvature bound for p ≥ 1.

Estimate of isodiametric constant for closed surfaces

We introduce a natural definition of L p -convergence of maps, p > 1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the L p -convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.

Concentration, Convergence, and Dissipation of Spaces

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist nitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with nite volume, of complex-dimension n 2. The group G is acting on the universal cover of M , which is isometric to H n . It is a CAT( 1) space of dimension 2n. The geometric dimension of G is 2n 1. We show that G does not act on any proper CAT(0) space of dimension 2n 1 properly by isometries. We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag-Solitar groups. AMS Classication 20F67; 20F65, 20F36, 57M20, 53C23

The Riemannian structure of Alexandrov spaces

We give an explicit estimate of the area of a closed surface by the diameter and a lower bound of curvature. This is better than Calabi–Cao’s (J Differ Geom 36(3): 517–549, 1992) estimate for a nonnegatively curved two-sphere.

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

We survey some parts of Gromov’s theory of metric measure spaces [6, Sect. 3.\(\frac{1}{2}\)], and report our recent works [14, 15, 16, 17], focusing on the asymptotic behavior of a sequence of spaces with unbounded dimension.